Emmanuel Breuillard - Laboratoire de Mathématiques, Bâtiment 425, Université Paris Sud 11, 91405 Orsay, France (email)
Abstract: This is an informal announcement of results to be described and proved in detail in . We give various results on the structure of approximate subgroups in linear groups such as $\SL_n(k)$. For example, generalizing a result of Helfgott (who handled the cases $n = 2$ and $3$), we show that any approximate subgroup of $\SL_n(\F_q)$ which generates the group must be either very small or else nearly all of $\SL_n(\F_q)$. The argument is valid for all Chevalley groups $G(\F_q)$. Extending work of Bourgain-Gamburd we also announce some applications to expanders, which will be proven in detail in .
Keywords: Approximate groups, growth, expander graphs.
Received: January 2010; Published: September 2010.